A224212 Number of nonnegative solutions to x^2 + y^2 <= n.

1, 3, 4, 4, 6, 8, 8, 8, 9, 11, 13, 13, 13, 15, 15, 15, 17, 19, 20, 20, 22, 22, 22, 22, 22, 26, 28, 28, 28, 30, 30, 30, 31, 31, 33, 33, 35, 37, 37, 37, 39, 41, 41, 41, 41, 43, 43, 43, 43, 45, 48, 48, 50, 52, 52, 52, 52, 52, 54, 54, 54, 56, 56, 56, 58, 62, 62, 62, 64

Offset: 0

Author

Alex Ratushnyak, Apr 01 2013

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000925 (first differences).

Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).

Cf. A057655 (number of integer solutions to x^2 + y^2 <= n).

Programs

Mathematica

nn = 68; t = Table[0, {nn}]; Do[d = x^2 + y^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

Python

for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        k+=1
  print(str(k), end=', ')

Formula

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^2. - Ilya Gutkovskiy, Mar 14 2017